A Polynomial Algorithm for Submap Isomorphism: Application to Searching Patterns in Images

Transcription

1 A Polynomial Algorithm for Submap Isomorphism: Application to Searching Patterns in Images Guillaume Damiand, Colin de la Higuera, Jean-Christophe Janodet, Emilie Samuel, Christine Solnon GbR 009

2 Motivations Search patterns in images Model images graphs (RAGs, Delaunay triangulation,...) Search patterns subgraph isomorphism NP-complete in the general case... but do we consider the right problem when graphs model images? Is there a subgraph isomorphism???

3 Motivations Search patterns in images Model images graphs (RAGs, Delaunay triangulation,...) Search patterns subgraph isomorphism NP-complete in the general case... but do we consider the right problem when graphs model images? Yes but... the two graphs look rather different! Graphs modeling images are planar and are embedded in planes. Let us compare planar embeddings of graphs!

4 Motivations Search patterns in images Model images graphs (RAGs, Delaunay triangulation,...) Search patterns subgraph isomorphism NP-complete in the general case... but do we consider the right problem when graphs model images? Yes but... the two graphs look rather different! Graphs modeling images are planar and are embedded in planes. Let us compare planar embeddings of graphs!

5 Overview of the talk 1 Motivations Combinatorial maps 3 Map isomorphism: definition and algorithm 4 Submap isomorphism: definition and algorithm 5 Back to plane graphs plane subgraph isomorphism 6 Experimental results 7 Further work

6 D Combinatorial maps (1/3) From plane graphs to D combinatorial maps Each edge is decomposed into linked darts Faces are defined by dart successions Plane graph Combinatorial map

7 D Combinatorial maps (/3) Definition [Lienhardt91] A D combinatorial map is defined by M = (D, β 1, β ) such that D = finite set of darts β 1 = permutation on D β = involution on D Notation: β 0 β β β

8 D Combinatorial maps (3/3) Open maps Some darts may be i-sewn with ɛ (ex.: 5, 6, 7, 14, 15, 10 et 1) By definition, 0 i, β i (ɛ) = ɛ Connected maps for every pair of darts (d, d ), there exists a path (d 1,..., d k ) st d 1 = d and d k = d 1 i < k, j i {0, 1, }, d i+1 = β ji (d i )

9 Overview of the talk 1 Motivations Combinatorial maps 3 Isomorphism of combinatorial maps 4 Submap isomorphism 5 Back to plane graphs 6 Experimental results 7 Further work

10 Map isomorphism Definition [Cori75, Lienhardt94] M = (D, β 1, β ) and M = (D, β 1, β ) are isomorphic if there exists a bijection f : D D such that d D, i {1, }, f (β i (d)) = β i (f (d)). function testisomorphism(m, M ) [Cori75] Input: connected open maps M and M Output: returns true iff M and M are isomorphic Choose d 0 D For each dart d 0 D do : f traverseandmatch(m, M, d 0, d 0 ) If f is an isomorphism function between M and M then return true return false

11 Map traversal and construction of f function traverseandmatch(m, M, d 0, d 0 ) Input: maps M and M and darts d D and d D Output: returns a matching f : D {epsilon} D {epsilon} For each dart d D do: f [d] nil f [d 0 ] d 0 Let S be an empty stack Push d 0 in S While S is not empty do f [ɛ] ɛ return f Pop a dart d from S For i {0, 1, } do If β i (d) ɛ and f [β i (d)] = nil then f [β i (d)] β i (f [d]) Push β i (d) in S

12 Complexity and correction Complexity in O( D ) At most D map traversals Each traversal is in O( D ) Correction If testisomorphism(m, M ) returns true, then M and M are isomorphic check that f is an isomorphism function before returning true If M and M are isomorphic then testisomorphism(m, M ) returns true the traversal is determined by the order β i are used to discover new darts

13 Overview of the talk 1 Motivations Combinatorial maps 3 Isomorphism of combinatorial maps 4 Submap isomorphism 5 Back to plane graphs 6 Experimental results 7 Further work

14 Submap isomorphism Definition M = (D, β 1, β ) is isomorphic to a submap of M = (D, β 1, β ) if there exists an injection f : D {ɛ} D {ɛ} such that f (ɛ) = ɛ and d D, i {1, } : if β i (d) ɛ then β i (f (d)) = f (β i(d)) if β i (d) = ɛ then, either β i (f (d)) = ɛ, or f 1 (β i (f (d))) is empty. a b r 1 M M M M is a submap of M M is not a submap of M because β (1) = β () = ɛ but β (f (1)) = f ()

15 Algorithm for submap isomorphism function testsubisomorphism(m, M ) Input: open connected maps M and M Output: returns true iff M is isomorphic to a submap of M Choose d 0 D For every dart d 0 D do : f traverseandmatch(m, M, d 0, d 0 ) If f is a subisomorphism function between M and M then return true return false Complexity in O( D D ) There are at most D map traversals Each traversal is in O( D )

16 Example with wrong initial darts 1 1 1

17 Example with wrong initial darts

18 Example with wrong initial darts

19 Example with wrong initial darts

20 Example with wrong initial darts All darts of the pattern are discovered end of TraverseAndMatch The matching isn t a subisomorphism testsubiso tries another dart

21 Example with good initial darts 1 1 1

22 Example with good initial darts

23 Example with good initial darts

24 Example with good initial darts

25 Example with good initial darts All darts of the pattern are discovered end of TraverseAndMatch The matching is a subisomorphism testsubiso returns true

26 Overview of the talk 1 Motivations Combinatorial maps 3 Isomorphism of combinatorial maps 4 Submap isomorphism 5 Back to plane graphs 6 Experimental results 7 Further work

27 From plane graphs to combinatorial maps (1/3)...or how to use submap isomorphism to solve some subgraph isomorphism problems... Compact plane subgraph isomorphism Plane graph embedding of a planar graph in a plane G 1 and G are plane-isomorphic if there exists a bijection f : N 1 N which preserves edges and topology G 1 is a compact plane subgraph of G if G 1 is plane isomorphic to a compact subgraph remove nodes and edges adjacent to the unbounded face Yes 5 No

28 From plane graphs to combinatorial maps (/3) Modeling a plane graph with a D map Associate a face in the map with every face of the graph Non plane isomorphic Non isomorphic maps?

29 From plane graphs to combinatorial maps (/3) Modeling a plane graph with a D map Associate a face in the map with every face of the graph Non plane isomorphic Non isomorphic maps

30 From plane graphs to combinatorial maps (/3) Modeling a plane graph with a D map Associate a face in the map with every face of the graph except the unbounded face open map w.r.t. β only (and only for one face) topological disk Non plane isomorphic Non isomorphic maps Non isomorphic maps

31 From plane graphs to combinatorial maps (3/3) Precondition for using test(sub)isomorphism(m, M ) M and M must be connected plane graphs must be connected......and their unbounded face must be bounded by an elementary cycle Yes 5 No a polynomial algorithm to solve compact plane subgraph isomorphism when unbounded faces are bounded by elementary cycles...

32 Overview of the talk 1 Motivations Combinatorial maps 3 Isomorphism of combinatorial maps 4 Submap isomorphism 5 Back to plane graphs 6 Experimental results 7 Further work

33 Experimental results (1/) Original image Combinatorial map of the segmented image (435 nodes, 4057 edges) => find the car in 60ms Corner points connected by Delaunay triangulation (140 nodes, 404 edges) => find the car in 10ms

34 Experimental results (/) Comparison with Vflib (induced subgraph isomorphism) g x x nodes randomly embedded in the plane + Delaunay triangulation sg i,j% subgraph of g i with j% of the nodes sg i,5% sg i,10% sg i,0% sg i,33% sg i,50% g i vf map vf map vf map vf map vf map g g g Vflib and map do not solve the same problem...but they (nearly always) find the same solutions

35 Overview of the talk 1 Motivations Combinatorial maps 3 Isomorphism of combinatorial maps 4 Submap isomorphism 5 Back to plane graphs 6 Experimental results 7 Further work

36 Further work Dealing with holes in subgraphs Solving subgraph isomorphism in polynomial time for graphs with a polynomial number of topologically different embeddings in n dimensional spaces identify such classes of graphs Mining frequent submaps Definition of Error tolerant similarity measures Maximum common submap Map edit distance integrate geometric and color information